Quantizations of generalized-Witt algebra and of Jacobson-Witt algebra in the modular case

We quantize the generalized-Witt algebra in characteristic 0 with its Lie bialgebra structures discovered by Song-Su (\cite{GY}). Via a modulo p reduction and a modulo "p-restrictedness" reduction process, we get 2^n{-}1 families of truncated p-polynomial noncocommutative deformations of the restricted universal enveloping algebra of the Jacobson-Witt algebra \mathbf{W}(n;\underline{1}) (for the Cartan type simple modular restricted Lie algebra of W type). They are new families of noncommutative and noncocommutative Hopf algebras of dimension p^{1+np^n} in characteristic p. Our results generalize a work of Grunspan (J. Algebra 280 (2004), 145--161, \cite{CG}) in rank n=1 case in characteristic 0. In the modular case, the argument for a refined version follows from the modular reduction approach (different from \cite{CG}) with some techniques from the modular Lie algebra theory.